Minimum number of double points over all immersed disks

Let $$K$$ be a knot in the boundary of a compact smooth 4-manifold $$X$$, and suppose that $$K$$ is the the kernel of $$\pi_1(\partial X) \to \pi_1(X)$$. Then $$K$$ is the boundary of some immersed disk $$D \to X$$ that has some fine number of double points. I am interested in knowing how to compute minimum number of such double points, minimized over all such immersed disks. For now, I will call it the immersion number of $$K$$ - denoted $$I(K)$$, for fun. Has this quantity been studied in the literature?

One thing to note about $$I(K)$$ is that it provides a lower bound on the minimum genus of an orientable surface bounding $$K$$ (the "4-ball genus" if $$X = B^4$$). I am interested in knowing examples of knots (even/especially with $$X = B^4$$) where $$I(K)$$ and the minimum genus of an orientable surface properly embedded in $$X$$ bounding $$K$$ are not equal.

• It is maybe not of your interest. In case when $K$ is not a knot but essential circle in $\partial X$ which is trivial in $X$ then we can find embedding of disk $D\to X$ which bounds our circle, right ? This is famous Dehn lemma in 3-manifolds but in 4-manifolds I don't know whether it holds. My interest is following: for given $M^3$ find minimal $N^4 \supset M$ with trivial first homology. Intuition is that $M$ should cut $N$ onto two pieces. I wonder when we can find $N$ also with trivial second homology which makes it homology sphere. – user21230 Oct 11 '18 at 14:17

For knots in $$S^3=\partial B^4$$ this is called the four-ball crossing number or clasp number.
For knots in $$S^3$$ you have the following inequalities relating $$I$$ to the $$4$$-ball genus $$g_4$$ and the unknotting number $$u$$
$$g_4\leq I \leq u.$$
• For reference, the knot $9_35$ has $u=3, I=2, g_4=1$ - it is the lowest crossing number knot with all of Marc's inequalities strict. – user101010 Oct 9 '18 at 19:30